Answer to Question #248136 in Discrete Mathematics for zid

Question

The heart beat rate has to be well regulated to provide enough oxygenated blood

throughout the body and so depends on feedback with the body’s oxygen demand. A simple discrete model of heart beat regulation is given by:

xt+1 = kxt(1 - xt)

Here xt represents the normalised heart beat rate at time t recorded once per minute. That is, the normalisation involves dividing the actual hear rate in beats per minute by 80 beats per minute.

The parameter k is a positive real number (hopefully) greater than 0.

(a) Assuming k = 1 what are the steady state solutions (also known as fixed points) for xt? That is, when xt+1 = xt

(b) Assuming k = 2 what are the fixed points?

(c) Write a MATLAB program using array operations to generate a table (with headings) of the normalised heart beat rate per minute starting at time t = 0 with the value of x0 entered by the user. Run your program with the maximum time set to 30 minutes. Show table and MATLAB code for x0 = 0.1 and k = 2

Answer 1

a)

$x_{t+1}=x_t$

$x_t=x_t(1-x_t)$

$x_t=0$

b)

$x_t=2x_t(1-x_t)$

$x_t=1/2$

c)

MATLAB code:

x=0.1;
beats=A(1,30); 
for k=1:30
A=2x*(1-x);
x=A;
disp(A)
end